# Message #267

From: Melinda Green <melinda@superliminal.com>

Subject: Re: [MC4D] other polychora and polytope permutation puzzles

Date: Sat, 03 Jun 2006 18:39:47 -0700

Roice Nelson wrote:

> I had the thought the other day that it would be cool to do a 4D

> version of the Megaminx. A little wikipedia and web searching

> revealed that the word ‘polychoron’ is a common (though not standard)

> word for a 4D ‘polytope’, which is the word for a generalized

> d-dimension polygon (a 3D polytope is a polyhedron). Also, I found

> that the 120-cell ( http://en.wikipedia.org/wiki/120-cell) is the

> polychoron that could be considered the 4D analog of the dodecahedron,

> which the Megaminx is based on.

Yep. Don and I have both done a bunch of work visualizing the 120 cell

and other regular 4D objects when we were in school together in 1985-86.

In roughly 1992 when we were working on probably our 2nd implementation

of MC4D, we talked a fair amount about puzzle versions of other

polytopes. Don began work on a general engine to produce puzzles from

general 4D polytopes but never finished it. It turns out that not all

regular polytopes slice up nicely into puzzles. If I recall correctly,

the good ones in N-D are those in which D faces meet at each vertex.

Those with more faces result in cuts that generate a bunch of little

"scrap" cells in addition to the normal ones. I think that we agree that

the 120-cell is the most beautiful of the regular 4D polytopes and we

dreamed about seeing its puzzle version.

>

> There is a really cool picture of an exploded 120-cell projected into

> 3D space about halfway down the page at

> http://home.inreach.com/rtowle/Polytopes/polytope.html. This

> is mostly what I was thinking about, though I imagined slightly

> different projection parameters that would have the closest face

> hidden like in MC4D.

I’m fairly sure that that image has the closest face hidden just like in

MC4D otherwise it would overlap the rest of the figure. IOW, it looks

like a pretty good projection for a usable puzzle to me. Note that they

also seem to have removed the next layer of cells which are flattened

even more than the thin ones you can see edge-on. I suppose they did

that to let you see deeper inside. You can see the wide, shallow

indentations where they would nest around the outside. A real 120-cell

with only the outer face removed would have only 12 outwardly facing

pentagons in the shape of a dodecahedron.

> I saw a picture having projection parameters I like at

> http://www.bathsheba.com/math/120cell/, though it isn’t a cool

> exploded view. The latter is more analogous to how I tried to flatten

> a 3D dodecahedron into 2D when thinking about this.

This view seems similar to your 5D puzzle in which the wireframe

rendering solves a good deal of the crowding problem. Perhaps a puzzle

version could have a slider that allowed you to quickly hide more or

fewer layers of cells either by making them invisible or turning them

into wireframe mode.

>

> Maybe the Magic120Cell puzzle would be too much with 120 faces. If

> they couldn’t realistically be distinguished by colors, stickers might

> have to be numbered.

I’m not completely sure about that but we’d have to test it to be

certain. Perhaps some highlighting tools would help to mitigate the problem.

> I have seen a Megaminx with only 6 colors which had colors repeated on

> opposite sides. Maybe doing something like that could help, though it

> does change the nature of the puzzle (aside: I wonder if repeating

> colors, but not on opposite sides, could produce a Megaminx that

> behaved exactly as the 12 colored one, since the 1-colored center

> pieces can’t change position and the other pieces have multiple colors

> to help distinguish where they belong. I’ll have to think more on

> that. If it was the case, maybe colors on a 120cell could even be

> repeated more than once without changing the puzzle behavior.?.)

Even if you can do that, it seems like it would make it harder to solve

than simply allowing 120 unique colors.

>

> Anyway, if the 120-cell is over the top, other polychora could make

> good puzzles.

Even though the display would be crowded, it may not be as bad as MC5D

and could have some advantages that might make it easier to solve than

the 4D cube for the same reasons that the Megaminx seems easier to solve

than the original Rubik’s cube due to the additional space in which you

can sequester some parts while working on others.

> The 4-simplex ( http://en.wikipedia.org/wiki/4-simplex) would be the

> 4D analog of Pyraminx, and would be easier and less screen-crazy than

> MC4D. That puzzle would also be interesting because as I imagine one

> projection of it, there would be no center face. There would be 4

> icosahedrons all pointing towards an empty center. The 5th

> icosahedron would be the hidden face closest to the viewer (think of

> the 3D to 2D case of projecting the icosahedron of a Pyraminx to 2D).

Icosahedral pieces? Is that right? I’d definitely love to try my hand at

solving this one since I can usually intuit my way to solving the 3D

version fairly quickly.

>

> An interesting thing about these puzzles is that relative to the

> orthogonal coordinate axes, face rotations can affect all coordinates

> of a sticker. Just like some face twists on Megaminx can change all 3

> coordinates of the moved stickers, some face twists on Magic120Cell or

> Magic4Simplex could change all 4 sticker coordinates. This is unlike

> the cube puzzles, and it seems this could make the programming more

> difficult, e.g. in MC4D you can reduce face rotations to 3D rotations

> by simply not dealing with 1 of the 4 coordinates, but this will no

> longer be possible. This wasn’t a problem in MC5D either, because we

> limited rotations to those parallel to coordinate planes and all face

> rotations only changed 2 coordinate values for stickers. I don’t

> understand it yet, but apparently 2 quaternions can represent any

> general 4D rotation (vs. 1 that is required for 3D rotations). What I

> don’t know is how to interpret the quaternion parameters with a

> geometric meaning, like the rotation plane and rotation angle. For a

> 3D rotation, the quaternion that represents it has a direct

> geometrical way to think about it in terms of an axis of rotation and

> an angle, but I’m not sure if the 4D case even has a similar

> geometrical interpretation. If anyone has any information on this, I

> would appreciate it.<br>
My understanding is that even though quaternions do generalize, the

property that makes them useful for managing 3D rotations is just a

quirk that does not generalize. OTOH, arbitrary N-dimensional rotation

matrices are not terribly difficult to deal with. Essentially all you

need to do is specify the particular 2D plane in which you want to

rotate and the amount of rotation you want and you can build the matrix

directly from that. Don is really good with all of this stuff and his

vec_h.c program can generate a C macro library with functions for doing

all of the matrix math in whatever dimension you want.

-Melinda